f(x)=x64-4x^2+k. Find the values of k so thath the zeros of f satisfy the specified charachteristics.

a.four real zeros
b.two real zeros, each of multiplicity of two
c.two real zeros and two complex zeros
d.four complez zeros

a.I don't know if this range goes for this or b. 1<x<3?
b. k=4 and x=+ √2 and x=- √2
And I don't know how to get c and d
any help would be appreciated :)

I gotta think you meant

f(x) = x^4-4x^2+k

we know that
x^4-4x^2 = x^2(x^2-4) = x^2(x-2)(x+2)
so it has 3 real roots, one repeated.

x^4-4x^2+4 = (x^2-4)^2 = (x-2)^2(x+2)^2
so it has 4 real roots (2 distinct)

x^4-4x^2+k has 4 distinct real roots if 0 < k < 4

x^4-4x^2+k has 4 complex roots if k>4

x^4-4x^2+k has two real, two complex roots if k < 0

To convince yourself of this, just check the graphs of the first two functions.

To find the values of k for which the zeros of f(x) satisfy the specified characteristics, let's analyze each case separately:

a) Four real zeros:
For f(x) to have four real zeros, the discriminant of the quadratic equation obtained by setting f(x) equal to zero must be greater than or equal to zero.
The quadratic equation is: x^4 - 4x^2 + k = 0.
We can use the discriminant formula, Δ = b^2 - 4ac, where a = 1, b = 0, and c = k.

Δ = 0^2 - 4(1)(k) = -4k.

Since Δ must be greater than or equal to zero, we have:
-4k ≥ 0
k ≤ 0

So, the values of k for which f(x) has four real zeros are k ≤ 0.

b) Two real zeros, each of multiplicity of two:
For f(x) to have two real zeros, each of multiplicity two, the discriminant must be equal to zero.
Using the same quadratic equation, Δ = -4k, we have:

Δ = 0
-4k = 0
k = 0

So, the value of k for which f(x) has two real zeros, each of multiplicity two, is k = 0. The zeros are x = +√2 and x = -√2.

c) Two real zeros and two complex zeros:
For f(x) to have two real zeros and two complex zeros, the discriminant must be less than zero.
Using the same quadratic equation, Δ = -4k, we have:

Δ < 0
-4k < 0
k > 0

So, the values of k for which f(x) has two real zeros and two complex zeros are k > 0.

d) Four complex zeros:
For f(x) to have four complex zeros, the discriminant must be less than zero. This is the same condition as in case c).

Therefore, the values of k for which f(x) has four complex zeros are also k > 0.

In summary:

a) k ≤ 0 for four real zeros.
b) k = 0 for two real zeros, each of multiplicity two (x = +√2 and x = -√2).
c) k > 0 for two real zeros and two complex zeros.
d) k > 0 for four complex zeros.

To find the values of k that satisfy the specified characteristics for the zeros of f(x) = x^4 - 4x^2 + k, we can use the discriminant of the quadratic equation. The discriminant can help us determine the nature of the roots.

a. Four real zeros:
For f(x) to have four real zeros, the discriminant of the quadratic equation obtained from f(x) should be positive.

The quadratic equation from f(x) is:
x^4 - 4x^2 + k = 0

Let's solve for the discriminant D:
D = (-4)^2 - 4 * 1 * k = 16 - 4k

For four real zeros, we need D > 0:
16 - 4k > 0
Simplifying, we have: -4k > -16
Dividing both sides by -4 and changing the direction of the inequality, we get: k < 4.

Therefore, for four real zeros, the values of k should be less than 4.

b. Two real zeros, each of multiplicity of two:
To have two real zeros, each of multiplicity of two, the discriminant D should be equal to zero.

Continuing from part a, we have D = 16 - 4k. Setting D = 0:
16 - 4k = 0
Simplifying, we find: k = 4

Therefore, for two real zeros, each of multiplicity of two, k should be equal to 4. The corresponding zeros are x = √2 and x = -√2.

c. Two real zeros and two complex zeros:
For two real zeros and two complex zeros, the discriminant D should be negative.

Continuing from part a, we have D = 16 - 4k. To find the range of k, we need D < 0:
16 - 4k < 0
Dividing both sides by -4 and changing the direction of the inequality, we get: k > 4.

Therefore, for two real zeros and two complex zeros, the values of k should be greater than 4.

d. Four complex zeros:
For four complex zeros, the discriminant D should be negative as well.

Continuing from part a, we have D = 16 - 4k. To find the range of k, we need D < 0:
16 - 4k < 0
Dividing both sides by -4 and changing the direction of the inequality, we get: k > 4.

Therefore, for four complex zeros, the values of k should be greater than 4.

To summarize:

a. Four real zeros: k < 4
b. Two real zeros, each of multiplicity of two: k = 4
c. Two real zeros and two complex zeros: k > 4
d. Four complex zeros: k > 4

For the given range 1 < x < 3, it is not directly related to the zeros of the function but rather specifies a range for the independent variable x. The above analysis was solely focused on finding the values of k based on the characteristics of the zeros.